Go to the content. | Move to the navigation | Go to the site search | Go to the menu | Contacts | Accessibility

| Create Account

Cecchinato, Nedda (2008) Bootstrap and approximation methods for long memory processes. [Ph.D. thesis]

Full text disponibile come:

Cover Image ( shown in the Abstract page)
Documento PDF

Abstract (english)

Fractionally integrated processes ARFIMA(p,d,q), introduced by Granger (1980) and Hosking (1981) independently, offer a useful tool to model the second order dependence structure (autocovariance and autocorrelation functions) of an observed time series. The literature is rich of paper on identification of the data generating process (dgp, from now on) and estimation of the parameters: Yajima (1985), Taqqu (1986) and Dahlhaus (1988), Dahlhaus (1989) wrote papers on parametric estimate of the memory parameter d, whereas Hurst (1951), Geweke and Porter-Hudak (1983), Higuchi (1988), Robinson (1995) and Hurvich (1998) developed semi-parametric estimation methods. It is not possible to define the best method, according to the situation each method offers advantages and drawbacks. Parametric estimators are asymptotically Normal and they are the most efficient, however in the case of misspecification of the model the estimates might be dramatically biased. On the other hand, semi-parametric estimators offer the possibility of estimating the long memory parameter from the short memory part with the drawback of a slower convergence rate (o(n -1/2)) or less) than with parametric techniques (o(n-1)). Moreover, Agiakloglou (1993) showed that the GPH Geweke and Porter-Hudak (1983) is biased in presence of arma parameter near the non-stationary area.

A generalisation of ARFIMA processes are the Gegenbauer processes, introduced by Hosking (1981) and then studied by Woodward (1989), Woodward (1998), Giraitis (1995), Smallwood (2004), Sadek (2004) and Caporale (2006). Also in this case parametric and semi-parametric technique are available in the literature. One the main problem is the maximization in a multidimensional space of the likelihood function because there is not a close form and the existing numerical procedures are quite burdensome. Semi-parametric procedures play and important role to compute good starting values to maximize the likelihood function and to identify the order of the dgp.

In the last years many bootstrap methods for time series have been developed, such as the model-based resampling, the block bootstrap (Kunsch, 1989), the autoregressive-aided periodogram bootstrap (Kreiss, 2003), the local bootstrap (Paparoditis, 1999), the sieve bootstrap (Kreiss, 1992), the parametric bootstrap (Andrews, 2006), the kernel bootstrap (Franke, 1992,Dahlhaus, 1996) and the phase scrambling (Theiler, 1992b). Bootstrap methods for time series have been widely used to build confidence intervals especially when asymptotic theory does not provide satisfactory results (Efron, 1979, Efron, 1982, Efron, 1987, Efron, 1987a, Hall, 1988, Hall, 1992, Arteche, 2005). The problem is still open when we want to replicate the dependence structure of a long memory process such as ARFIMA(p,d,q).

In this thesis we develop a new bootstrap method for time series, the ACF bootstrap, based on a result of Ramsey (1974), that generates the surrogate series from the observed autocorrelation function. The thesis is divided in five chapters: the first two chapters review some literature, the last three chapters are new contributions.

The first chapter reviews the literature on long memory processes, the properties of their sample autocorrelation and autocovariance functions, the most common parametric and semi-parametric estimators and, shortly, Gegenbauer processes. In the second chapter, we introduce briefly some bootstrap methods for time series.

In Chapter 3 we introduce the new bootstrap method. We apply the ACF bootstrap to improve the performance of semi-parametric estimators for the memory parameter d for ARFIMA(0,d,0) processes in terms of smaller standard error, smaller mean squared error and better coverage for confidence intervals. Since the condition of Gaussianity of the observed process is very restrictive, we show, by means of Monte Carlo simulation, that the method is consistent even relaxing this hypothesis. In particular the method seems to be robust against fat tails and itshape asymmetry}. Another application is building confidence intervals for the memory parameter d. For the parametric Whittle estimator, the confidence intervals based on the bootstrap distribution have a closer coverage to the theoretical level if the time series is relatively short (n=128,300). For semi-parametric estimators, applying bootstrap improves coverage of confidence intervals for d when the dgp is a ARFIMA(1,d,0) process.

In Chapter 4 we study the asymptotic behaviour of sample autocovariance and autocorrelation functions of a long memory processes. This results are useful to give a theoretical support for the consistency of the method in replicating long memory.

Last, in Chapter 5, we propose an algorithm to estimate non-parametrically the parameters of a Gegenbauer process with one and two peaks in the spectral density. The bootstrap method will be useful to give an estimate of the distribution of the frequency parameter eta. Its asymptotic distribution is given for the estimators proposed by Chung (1996) and Sadek (2004) but it is very complicate and difficult to handle. The main aim is proposing a method to identify seasonal persistences and provide starting values for maximize a (penalized) likelihood function.
[brace not closed]

Statistiche Download - Aggiungi a RefWorks
EPrint type:Ph.D. thesis
Tutor:Bordignon, Silvano
Supervisor:Bisaglia, Luisa and Wolff, Rodney
Ph.D. course:Ciclo 19 > Corsi per il 19simo ciclo > STATISTICA
Data di deposito della tesi:2008
Anno di Pubblicazione:2008
Key Words:bootstrap, long memory, time series, Edgeworth series, Gegenbauer processes
Settori scientifico-disciplinari MIUR:Area 13 - Scienze economiche e statistiche > SECS-S/01 Statistica
Area 13 - Scienze economiche e statistiche > SECS-S/03 Statistica economica
Struttura di riferimento:Dipartimenti > Dipartimento di Scienze Statistiche
Codice ID:552
Depositato il:12 Sep 2008
Simple Metadata
Full Metadata
EndNote Format


I riferimenti della bibliografia possono essere cercati con Cerca la citazione di AIRE, copiando il titolo dell'articolo (o del libro) e la rivista (se presente) nei campi appositi di "Cerca la Citazione di AIRE".
Le url contenute in alcuni riferimenti sono raggiungibili cliccando sul link alla fine della citazione (Vai!) e tramite Google (Ricerca con Google). Il risultato dipende dalla formattazione della citazione.

1. Abramowitz, M. and I. Stegun (Eds.) (1972). Handbook Of Mathematical Functions With Formulas, Graphs, And Mathematical Tables. New York: Dover. Cerca con Google

2. Agiakloglou, C., P. Newbold, and M. Wohar (1993). Bias in an estimator of the fractional difference parameter. Journal of Time Series Analysis 14 (3), 235-246. Cerca con Google

3. Akaike, H. (1974). A new look to the statistical model identification. IEEE Trans. Autom. Control 19, 716-723. Cerca con Google

4. Albin, J. (1998a). A note on rosenblatt distributions. Statistics & Probability Letters 40 (1), 83-91. Cerca con Google

5. Albin, J. (1998b). On extremal theory for self-similarity processes. The Annals of Probability 26 (2), 743-793. Cerca con Google

6. Anderson, T. and A.Walker (1964). On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. Annals of Mathematical Statistics 35 (3), 1296-1303. Cerca con Google

7. Andrews, D., O. Lieberman, and V. Marmer (2006). Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes. Journal of Econometrics 133, 633-702. Cerca con Google

8. Arteche, J. and J. Orbe (2005). Bootstrapping the log-periodogram regression. Economics Letters 86, 79-85. Cerca con Google

9. Arteche, J. and P. Robinson (2000). Semiparametric inference in seasonal and cyclical long memory processes. Journal of Time Series Analysis 21 (1), 1-25. Cerca con Google

10. Baillie, R. (1996). Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 5-59. Cerca con Google

11. Barndorff-Nielsen, O. and D. Cox (1989). Asymptotic techniques for use in statistics. Chapman & Hall. Cerca con Google

12. Barnett, A. (2002). On the use of bispectrum to detect and model non-linearity. Ph. D. thesis, School of Physical Science, The University of Queensland. Cerca con Google

13. Bartlett, M. S. (1946). On the theoretical speci_cation and sampling properties of autocorrelated time-series. Supplement of the Journal of the Royal Statistical Society 8 (1), 27-41. Cerca con Google

14. Beran, J. (1994). Statistics for Long-Memory Processes. Chapmann & Hall. Cerca con Google

15. Bhattacharya, R., V. Gupta, and W. Waymire (1983). The Hurst e_ect under trends. Journal of Applied Probability 20, 649-662. Cerca con Google

16. Bickel, P. and P. Bühlmann (1999). A new mixing notion and functional central limit theorems for a SIEVE bootstrap in time series. Bernoulli 5 (3), 413-446. Cerca con Google

17. Bickel, P. and D. Freedman (1981). Some asymptotic theory for the bootstrap. The Annals of Statistics 9 (6), 1196-1217. Cerca con Google

18. Bisaglia, L. and D. Guègan (1998). A comparison of techniques of estimation in longmemory process. Computational statistics and data analysis 27, 61-81. Cerca con Google

19. Bose, A. (1988). Edgeworth correction by bootstrap in autoregressions. The Annals of Statistics 16 (4), 1709-1722. Cerca con Google

20. Box, G. and G. Jenkins (1976). Time Series Analysis Forecasting and Control. 2nd edition. San Francisco: Holden-Day. Cerca con Google

21. Braun, W. and R. Kulperger (1997). Properties of a Fourier bootstrap method for time series. Communications in Statistics Theory and Method 26 (6), 1329-1336. Cerca con Google

22. Brockwell, P. and R. Davis (1991). Time series: theory and methods. Springer-Verlag. Cerca con Google

23. Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli 3, 123-148. Cerca con Google

24. Bühlmann, P. (1998). Sieve bootstrap for smoothing in nonstationary time series. The Annals of Statistics 26 (1), 48-82. Cerca con Google

25. Bühlmann, P. (2002). Bootstrap for time series. Statistical science 17 (1), 52-72. Cerca con Google

26. Burlaga, L. and L. Klein (1986). Fractal structure of the interplanetary magnetic field. Journal of Geophysical Research 91, 347-350. Cerca con Google

27. Caporale, G. and L. Gil-Alana (2006). Long memory at the long-run and the seasonal monthly frequencies in the US money stock. Applied Economics Letters 13, 965-968. Cerca con Google

28. Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary sequence. The Annals of Statistics 14 (3), 1171-1179. Cerca con Google

29. Cavazos-Cadena, R. (1994). The asymptotic distribution of sample autocorrelations for a class of linear filters. Journal of Multivariate Analysis 48, 249-274. Cerca con Google

30. Chatfield, C. (1996). The analysis of time series: an introduction. Chapman & Hall. Cerca con Google

31. Chung, C.-F. (1996). Estimating a generalized long memory process. Journal of Econometrics 73 (1), 237-259. Cerca con Google

32. Crilly, A., R. Earnshaw, and H. Jones (Eds.) (1991). Fractals and Chaos. Springer-Verlag. Cerca con Google

33. Dahlhaus, R. (1988). Small sample e_ects in time series analysis: a new asymptotic theory and a new estimate. The Annals of Statistics 16 (2), 808-841. Cerca con Google

34. Dahlhaus, R. (1989). E_cient parameter estimation for self-similar processes. Annals of Statistics 17, 1749-1766. Cerca con Google

35. Dahlhaus, R. and D. Janas (1996). A frequency domain bootstrap for ratio statistics in time series analysis. The Annals of Statistics 24 (5), 1934-1963. Cerca con Google

36. Datta, S. and T. Sriram (1997). A modi_ed bootstrap for autoregression without stationarity. Journal of Statistical Planning and Inference 59, 19-30. Cerca con Google

37. Davison, A. and D. Hinkley (1997). Bootstrap methods and their application. Cambridge university press. Cerca con Google

38. Devaney, R. (1999). The mandelbrot set, the farey tree, and the fibonacci sequence. The American Mathematical Monthly 106 (4), 289-302. Cerca con Google

39. Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics 7, 1-26. Cerca con Google

40. Efron, B. (1982). The jackknife, the bootstrap and other resampling plans. CBMS-NSF Regional conference series in applied mathematics. Cerca con Google

41. Efron, B. (1987a). Better bootstrap confidence intervals. Journal of the American Statistical Association 82 (397), 171-185. Cerca con Google

42. Efron, B. (1987b). Better bootstrap confidence intervals: rejoinder. Journal of the American Statistical Association 82 (397), 198-200. Cerca con Google

43. Efron, B. (1987c). Empirical confidence intervals based on bootstrap samples: comment. Journal of the American Statistical Association 82 (399), 754. Cerca con Google

44. Efron, B. and R. J. Tibshirani (1993). An Introduction to the Bootstrap. Chapman & Hall. Cerca con Google

45. Fox, R. and M. Taqqu (1986). Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series. The Annals of Statistics 14 (2), 517-532. Cerca con Google

46. Franco, G. and V. Reisen (2007). Bootstrap approaches and confidence intervals for stationary and non-stationary long-range dependence. Physica A 375 (2), 546-562. Cerca con Google

47. Franke, J. and W. Härdle (1992). On bootstrapping kernel spectral estimates. The Annals of Statistics 20 (1), 121-145. Cerca con Google

48. Geweke, J. and S. Porter-Hudak (1983). The estimation and application of long memory time series model. Journal of Time Series Analysis 4 (4), 221-238. Cerca con Google

49. Giraitis, L., P. Kokoszka, and R. Leipus (2001). Testing for long memory in the presence of a general trend. Journal of Applied Probability 38 (4), 1033-1054. Cerca con Google

50. Giraitis, L. and R. Leipus (1995). A generalized fractionally differencing approach in longmemory Cerca con Google

51. modeling. Lithuanian Mathematical Journal 35 (1), 53-65. Cerca con Google

52. Giraitis, L., P. Robinson, and D. Surgailis (1999). Variance-type estimation of long memory. Stochastic Processes and their Applications 80, 1-24. Cerca con Google

53. Giraitis, L. and D. Surgailis (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate. Probability Theory and Related Fields 86, 87-104. Cerca con Google

54. Granger, C. and R. Joyeux (1980). An introduction to long memory time series models and fractional di_erencing. Journal of Time Series Analysis 1 (1), 15-29. Cerca con Google

55. Gray, H., N. Zhang, and W. Woodward (1994). Correction on 'On generalized fractional processes'. Journal of Time Series Analysis 15, 561-562. Cerca con Google

56. Hall, P. (1988). Theoretical comparison of bootstrap confidence interval. The Annals of Statistics 16 (3), 927-953. Cerca con Google

57. Hall, P. (1992a). The Bootstrap and Edgeworth Expansion. Springer-Verlag. Cerca con Google

58. Hall, P. (1992b). Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. The Annals of Statistics 20 (2), 675-694. Cerca con Google

59. Hall, P. and C. Heyde (1980). Martingale limit theory and its application. New York: Academic Press. Cerca con Google

60. Hannan, E. (1973). The asymptotic theory of linear time series models. Journal of Applied Probability 10, 130-145. Cerca con Google

61. He, S. (1996). A note on the asymptotic normality of sample autocorrelations for a linear stationarity sequence. Journal of Multivariate Analysis 58, 182-188. Cerca con Google

62. Higuchi, T. (1988). Approach to an irregular time series on the basis of the fractal theory. Physica D 31, 277-283. Cerca con Google

63. Hinkley, D. (1988). Bootstrap methods (with discussion). Journal of the Royal Statistical Society, B 50, 321-337. Cerca con Google

64. Hosking, J. (1981). Fractional di_erencing. Biometrika 68 (1), 165-176. Cerca con Google

65. Hosking, J. (1996). Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series. Journal of Econometrics 73, 261-284. Cerca con Google

66. Hurst, H. (1951). Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116, 770-799. Cerca con Google

67. Hurvich, C., R. Deo, and J. Brodsky (1998). The mena squared error of Geweke and Porter- Hudak's estimator of the memory parameter of a long-memory time series. Journal of Time Series Analysis 19 (1), 19-46. Cerca con Google

68. Kapetanios, G. and Z. Psaradakis (2006). Sieve bootstrap for strongly dependent stationary processes. Working Paper No. 552, Queen Mary, University of London. Cerca con Google

69. Kreiss, J. (1992). Bootstrap procedures for AR(1)-process. Springer: Heidelberg. Cerca con Google

70. Kreiss, J.-P. and E. Paparoditis (2003). Autoregressive-aided periodogram bootstrap for time series. Institute of Mathematical Statistics 31 (6), 1923-1955. Cerca con Google

71. Künsch, H. (1989). The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17 (3), 1217_1241. Cerca con Google

72. Künsch, H. R. (1987). Statistical aspects of self-similar processes. Proceedings of the First World Congress of the Bernoulli Society 1, 67-74. Cerca con Google

73. Lahiri, S. (1992). Resampling Methods for Dependent Data. Springer. Cerca con Google

74. Leland, W., M. Taqqu, W. Willinger, and D. Wilson (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Network 2 (1), 1-15. Cerca con Google

75. Leonenko, N. and V. Anh (2001). Rate of convergence to the Rosenblatt distribution fro additive functionals of stochastic processes with long-range dependence. Journal of Applied Mathematics and Stochastic Analysis 14 (1), 27-46. Cerca con Google

76. Li, H. and G. Maddala (1996). Bootstrapping time series models. Econometric reviews 15 (2), 115-158. Cerca con Google

77. Mandelbrot, B. (1975). Limit theorems of the self-normalized range for weakly and strongly dependent processes. Z. Wahr. verw. Geb. 31, 271-285. Cerca con Google

78. Mandelbrot, B. and J. van Ness (1968). Fractional brownian motion, fractional noise and application. SIAM Review 10, 422-437. Cerca con Google

79. Newbold, P. and C. Agiakloglou (1993). Bias in the sample autocorrelations of fractional noise. Biometrika 80 (3), 698-702. Cerca con Google

80. Nur, D., R. Wolff, and K. Mergensen (2001). Phase randomisation: numerical study of higher order cumulants behaviour. Computational Statistics & Data Analysis 37, 487-513. Cerca con Google

81. Palma, W. and N. Chan (2005). E_cient estimation of seasonal long-range-dependent processes. Journal of Time Series Analysis 26 (6), 863-892. Cerca con Google

82. Paparoditis, E. and D. Politis (1999). The local bootstrap for periodogram statistics. Journal of Time Series Analysis 20 (2), 193-222. Cerca con Google

83. Paparoditis, E. and D. Politis (2002). The local bootstrap for Markov processes. Journal of Statistical Planning and Inference 108, 301-328. Cerca con Google

84. Pesaran, M. (1974). On the general problem of model selection. Review of Economic Studies 41, 153-171. Cerca con Google

85. Plasmans, J. (2006). Modern linear and nonlinear econometrics. Springer. Cerca con Google

86. Politis, D. (2003). The impact of bootstrap methods on time series analysis. Statistical science 18 (2), 219-230. Cerca con Google

87. Politis, D. and J. Romano (1994). The stationary bootstrap. JASA 89 (428), 1303-1313. Cerca con Google

88. Poskitt, D. (2007). Properties of the Sieve bootstrap for fractionally integrated and noninvertible processes. Journal of Time Series Analysis Accepted. Cerca con Google

89. Priestley, M. (1988). Spectral Analysis and Time Series. Harcourt Brace & Company, Publisher. Cerca con Google

90. Ramsey, F. (1974). Characterization of the partial autocorrelation function. Annals of Statistics 2, 1296-1301. Cerca con Google

91. Reisen, V., A. Rodrigues, and W. Palma (2006). Estimating seasonal long-memory processes: a Monte Carlo study. Journal of Statistical Computation and Simulation 76 (4), 305-316. Cerca con Google

92. Revuz, D. and M. Yor (1994). Continuous Martingales and Brownian Motion. Springer- Verlag, Second Edition. Cerca con Google

93. Robinson, P. (1995a). Gaussian semiparametric estimation of long range dependence. The Annals of Statistics 23 (5), 1630-1661. Cerca con Google

94. Robinson, P. (1995b). Log-periodogram regression of time series with long range dependence. The Annals of Statistics 23 (3), 1048-1072. Cerca con Google

95. Rogers, L. and D. Williams (1979). Diffusions, Markov processes, and martingales. Volume One: Foundations. John Wiley & Sons. Cerca con Google

96. Sadek, N. and A. Khotanzad (2004). K-Factor Gegenbauer ARMA process for network traffic simulation. Proceedings of the Ninth International Symposium on Computers and Communications 2004 2, 963-968. Cerca con Google

97. Samorodnitsky, G. and M. Taqqu (1994). Stable non-Gaussian random processes. Chapman & Hall. Cerca con Google

98. Shao, J. and D. Tu (1995). The Jackknife and Bootstrap. Springer. Cerca con Google

99. Silva, E., G. Franco, V. Reisen, and F. Cruz (2006). Local bootstrap approaches for fractional di_erential parameter estimation in ARFIMA models. Computational statistics & data analysis 51 (2), 1002-1011. Cerca con Google

100. Smallwood, A. and P. Beaumont (2004). Multiple frequency long memory. Discussion paper, Department of Economics, University of Oklahoma. Cerca con Google

101. Sowell, F. (1992). Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of Econometrics 53, 165-188. Cerca con Google

102. Swanepoel, J. and J. van Wyk (1986). The bootstrap applied to power spectral density function estimation. Biometrika 73 (1), 135-141. Cerca con Google

103. Taqqu, M., V. Teverovsky, and W. Willinger (1995). Estimators for long-range dependence: an empirical study. Fractals 3, 785-798. Cerca con Google

104. Teverovsky, V. and M. Taqqu (1997). Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator. Journal of Time Series Analysis 18 (3), 279-304. Cerca con Google

105. Theiler, J., S. Eubank, A. Longtin, B. Galdrikian, and J. Farmer (1992). Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 77-94. Cerca con Google

106. Theiler, J. and D. Prichard (1996). Constrained-realization Monte-Carlo for hypothesis testing. Physica D 58, 221-235. Cerca con Google

107. Tudor, C. (2006). Analysis of the Rosenblatt process. Working paper, www.archivesouvertes.fr . Vai! Cerca con Google

108. Wei, W. (1990). Time series analysis: univariate and multivariate methods. Addison- Wesley Publishing Company. Cerca con Google

109. Whittle, P. (1951). Hypothesis testing in time series analysis. Hafner, New York. Cerca con Google

110. Woodward, W., Q. Cheng, and H. Gray (1998). A k-factor GARMA long memory model. Journal of time series analysis 19 (4), 485-504. Cerca con Google

111. Woodward, W., H. Gray, and N. Zhang (1989). On generalized fractional process. Journal of time series analysis 10, 233-257. Cerca con Google

112. Yajima, Y. (1985). On estimation of long-memory time series models. Australian Journal of Statistics 27, 303-320. Cerca con Google

Download statistics

Solo per lo Staff dell Archivio: Modifica questo record